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Digital Filtering using Pseudo Fermat Transforms

Publishing Venue

IBM

Related People

Nussbaumer, H: AUTHOR

Abstract

With the growing number of digital filtering applications, efficient implementation of digital filters is a matter of increasing importance. Discussed here is the use of some finite field transforms as a way to improve efficiency in digital filter computation.

Country

United States

Language

English (United States)

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Digital Filtering using Pseudo Fermat Transforms

With the growing number of digital filtering applications, efficient
implementation of digital filters is a matter of increasing importance. Discussed
here is the use of some finite field transforms as a way to improve efficiency in
digital filter computation.

Direct computation of a nonrecursive digital filtering process corresponds to a
continuous convolution, and requires N multiplications and N-1 additions per
output sample for an N-taps filter. This processing workload can be drastically
reduced by computing the convolution with discrete Fourier transforms (DFT).

In such an approach, the continuous convolution is transformed into a series
of circular convolutions by dividing the input sequence into blocks, to which a
suitable number of 0's are appended as in the conventional overlap-add, overlap-
save approaches. The circular convolutions can then be computed by using
discrete Fourier transforms evaluated via a fast Fourier transform algorithm
(FFT).

With this technique, the bulk of the processing workload corresponds to
evaluating the fourier transforms and is proportional to N log(2) N instead of N(2).

Recently, Rader in IEEE Transactions on Computers C21-1269 (1972),
Agarwal and Burrus in IEEE Transactions on Acoustics, Speech and Signal
Processing ASS P 22-87 (1974), and in Proceedings IEEE 63-550 (1975), have
proposed to replace Fourier transforms by Mersenne and Fermat transforms to
implement digital filters. These two transforms, which have the convolution
property, can be computed without multiplications and are, therefore, potentially
much more efficient than the DFT for the evaluation of convolutions.

Their main drawback is that, because all operations are performed in a finite
ring of integers with arithmetic carried out modulo p, there is a rigid relationship
between transform length and word length. Moreover, and particularly in the
case of the Fermat transform, there is only a very limited choice of possible word
lengths so that a significant amount of computing power may be wasted, by
operating on word lengths much longer than would be required to achieve a
given precision on the final result.

Defined here is a complex pseudo Fermat transform which is well adapted for
filtering complex signals, and allows increased transforms and convolution
lengths when compared to conventional Fermat transforms.